When $\displaystyle {q}$ thousand items of a new product have been sold, the total profits $\displaystyle {P}$ in thousands of dollars is approximated by:

$\displaystyle {P}={q}{\left(-{3}{q}+{28}\right)}-{\left({42}+{2}{q}\right)}$

A) Suppose 14 thousand dollars of profits have been earned. (It is possible to have negative profits in this context. This means the item is not yet profitable.) Find a quadratic equation in the form $\displaystyle {a}{q}^{{2}}+{b}{q}+{c}={0}$ you can solve to determine the number of items sold so far. Don't forget that $\displaystyle {q}$ is your variable.

Answer: The equation is .
(Use the variable $\displaystyle {q}$.)
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B) Solve your equation in part [a] by factoring and use your answer to indicate the smallest number of items that need to be sold in order to attain a profit of 14 thousand dollars.

Answer: To attain profits of 14 thousand dollars, thousand items must be sold. (And this is the smallest number of these items needed to attain that profit level.)